Randomly generated polytopes for testing mathematical programming algorithms

Randomly generated polytopes are used frequently to test and compare algorithms for a variety of mathematical programming problems. These polytopes are constructed by generating linear inequality constraints with coefficients drawn independently from a distribution such as the uniform or the normal. It is noted that this class of 'random' polytopes has a special property: the angles between the hyperplanes, though dependent on the specific distribution used, tend to be equal when the dimension of the space increases. Obviously this structure of 'random' polytopes may bias test results

Random polytopes: Their definition, generation and aggregate properties

The definition of random polytope adopted in this paper restricts consideration to those probability measures satisfying two properties. First, the measure must induce an absolutely continuous distribution over the positions of the bounding hyperplanes of the random polytope; and second, it must result in every point in the space being equally as likely as any other point of lying within the random polytope. An efficient Monte Carlo method for their computer generation is presented together with analytical formulas characterizing their aggregate properties. In particular, it is shown that the expected number of extreme points for such random polytopes increases monotonically in the number of constraints to the limiting case of a polytope topologically equivalent to a hypercube. The implied upper bound of 2 n where n is the dimensionality of the space is significantly less than McMullen's attainable bound on the maximal number of vertices even for a moderate number of constraints.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/47911/1/10107_2005_Article_BF01585093.pd

Minimum Length Embedding of Planar Graphs at Fixed Vertex Locations

We consider the problem of finding a planar embedding of a graph at fixed vertex locations that minimizes the total edge length. The problem is known to be NP-hard. We give polynomial time algorithms achieving an O(n√logn) approximation for paths and matchings, and an O(n) approximation for general graphs

Kursvärderingar som underlag för kvalitetsutveckling

I detta bidrag tas utgångspunkt i hur framtagningen av några policydokument för kursvärderingsarbete kännetecknats av ett gränsöverskridande samarbete, samlade erfarenheter och aktiva studenter. En översyn av hur ,genomförandet lyckats och i vilken utsträckning kursvärderingar ligger till grund för kvalitetsutvecklingsarbetet har undersökts. Exemplet kommer från Högskolan Kristianstad

Multi-messenger Observations of a Binary Neutron Star Merger

International audienceOn 2017 August 17 a binary neutron star coalescence candidate (later designated GW170817) with merger time 12:41:04 UTC was observed through gravitational waves by the Advanced LIGO and Advanced Virgo detectors. The Fermi Gamma-ray Burst Monitor independently detected a gamma-ray burst (GRB 170817A) with a time delay of $\sim 1.7\,{\rm{s}}$ with respect to the merger time. From the gravitational-wave signal, the source was initially localized to a sky region of 31 deg(2) at a luminosity distance of ${40}_{-8}^{+8}$ Mpc and with component masses consistent with neutron stars. The component masses were later measured to be in the range 0.86 to 2.26 $\,{M}_{\odot }$. An extensive observing campaign was launched across the electromagnetic spectrum leading to the discovery of a bright optical transient (SSS17a, now with the IAU identification of AT 2017gfo) in NGC 4993 (at $\sim 40\,{\rm{Mpc}}$) less than 11 hours after the merger by the One-Meter, Two Hemisphere (1M2H) team using the 1 m Swope Telescope. The optical transient was independently detected by multiple teams within an hour. Subsequent observations targeted the object and its environment. Early ultraviolet observations revealed a blue transient that faded within 48 hours. Optical and infrared observations showed a redward evolution over ∼10 days. Following early non-detections, X-ray and radio emission were discovered at the transient’s position $\sim 9$ and $\sim 16$ days, respectively, after the merger. Both the X-ray and radio emission likely arise from a physical process that is distinct from the one that generates the UV/optical/near-infrared emission. No ultra-high-energy gamma-rays and no neutrino candidates consistent with the source were found in follow-up searches. These observations support the hypothesis that GW170817 was produced by the merger of two neutron stars in NGC 4993 followed by a short gamma-ray burst (GRB 170817A) and a kilonova/macronova powered by the radioactive decay of r-process nuclei synthesized in the ejecta